62Well this was a gr8 question by Abhishek....
I got blown as well...
I thought it was 2 but it is actually 3.. the graph method killed me :D
On carefullly seeing i realised my graph was wrong :(
There is one root between 4 and 5
!!
How do we find out?
I cannot think of a brilliant solution...
But yeah u could try taking derivative.. find where it is zero...
check the signs at these points and +- infinity.
62
Lokesh Verma
·2008-10-19 23:27:53
finding the zeroes will need graphs. That is probably the best way!
Unfortunately, even here, we cant get a exact value of roots!
I am not sure how a better and more correct solution could be found!!!
Ne suggestions?
1now..........
how to know when graph method will be correct n when it will betray us this way....??????????
33ok skygirl two soln u figured 0 and 1 well that was the trick and u did it now at x=1 x2+1 overtook 2x and lim(x→∞) 2x-(x2+1)>0 so 2x overtakes x2+1 so it will definitely cut x2+1 giving a third soln. but will there be two more solns it is tricky (definitely it will have odd no of solns)
In objective question this much info is enough for choosing a option this will never be asked in subjective.
I solved it in test options given were
1
2
3
4
now which option is odd and i knew it will be more than 2 so 3 solns
Further U can check behaviour by putting some values. It is helpful in tests..
1Can you prove lim (x→∞) 2x - (x2+1) > 0
I'm not really convinced . expand 2x . That'll give .
1+(xln2) + (xln2)2 ...... -x2 -1 .
= xln2 + x2 ( .693 2 - 1) ....... I mean this is infinity you're dealing with. So , I just don't feel you can state that.
62
Lokesh Verma
·2008-10-20 13:07:03
nothing to prove srinath....
actually exponentials grow faster than polynomials in the long run....
see the derivative...
2xln2 and 2x
as x grows.. the first one far far supersedes the 2nd... it is inevitable that it will go beyond the 2nd
From now on, take it for a fact :)
1
voldy
·2008-10-20 13:14:03
the derivative gives the slope of the tangent , so it means that since 2xln2 increases more , the fn achieves greater values for small increments of x . This is what you mean . Right?
Thank you.
62
Lokesh Verma
·2008-10-20 13:18:24
yes i mean that..
so essentially one is moving very very fast.. and forever... so it will overtake the slower one evenutally :)
